In this section we give an extension of the Mabuchi functional to the setting of GK structures of symplectic type on a Kähler background. To begin we first observe that it is possible to canonically associate a Kähler metric to every symplectic-type GK structure on a Kähler manifold. First, using Hodge decomposition, the deRham class \(\alpha = [F]\)  defines an element \(\overline{\alpha }\in H^{1,1}(M, {\mathbb {R}})\) . Using that \(F\)  tames \(J\)  and a result of Demailly-Paun [1]}, it follows that \(\overline{\alpha }\)  is a Kähler class of \((M, J)\) . By Yau's theorem [2]}, we can find a unique Kähler metric \(\overline{\omega }\in \overline{\alpha }\)  with the property
\(\overline{\omega }^{[m]} = c F^{[m]}, \qquad c =\frac{\overline{\alpha }^{[m]}}{[F]^{[m]}}.\) 
